Showing papers by "Andrei V. Kelarev published in 1994"

Combinatorial properties and homomorphisms of semigroups

Abstract: A complete description of ultrarepetitive semigroups is given. As an application of this result all semigroups S are described such that the class of permutational semigroups is S-closed, i.e., closed under taking preimages of certain homomorphisms onto S.

Nil properties for rings which are sums of their additive subgroups

Abstract: Suppose that a ring R is a direct sum of a finite number of its additive subgroups, and the union of these subgroups is closed under multiplication. We show that if all rings among these subgroups are nilpotent (left T-nilpotent, locally nilpotent or Baer radical), then the whole ring R satisfies the same property.

Homogeneity of the radical of semigroup-graded rings

Abstract: We show that cancellativity is a necessary condition on a semigroup S for the Jacobson radical of all S-graded rings to be homogeneous. We introduce some related classes of semigroups S for which the Jacobson radical of certain S-graded rings is always homogeneous. Commutative and regular semigroups in these classes are described

Finiteness conditions for special band graded rings

Abstract: A number of chain (initeness conditions arc well-known classical concepts of ring theory. Properlies of this soi l have been investigated by many authors for various ring constructions, in particular, for semigroup rings (cf. [25]). The aim of the present paper is to carry out analogous investigations for special band graded rings, which arc closely related to semigroup rings. Namely, we shall deal with right Artinian, right Noetherian, semilocal, local, scalarly local, right perfect and scmiprimary rings.

On the nilpotency of the Jacobson radical of semigroup rings

Abstract: Munn [11] proved that the Jacobson radical of a commutative semigroup ring is nil provided that the radical of the coefficient ring is nil. This was generalized, for semigroup algebras satisfying polynomial identities, by Okninski [14] (cf. [15, Chapter 21]), and for semigroup rings of commutative semigroups with Noetherian rings of coefficients, by Jespers [4]. It would be interesting to obtain similar results concerning rings with nilpotent Jacobson radical. For band rings this was accomplished in [12], and for special band-graded rings in [13, §6]. However, for commutative semigroup rings analogous implication concerning the nilpotency of the radicals is not true: it follows from [7, Theorems 44.1 and 44.2], that if F is a field with charF = p and G is an infinite abelian p-group, then the Jacobson radical J(FG) is nil but not nilpotent. On the other hand, Braun [1] proved that the Jacobson radical of every finitely generated PI-algebra over a Noetherian ring is nilpotent. This famous result has several important corollaries (cf. [9], [19]). It shows that the existence of a finite generating set is a natural condition which may influence the nilpotency of the Jacobson radical of a ring. We shall prove the following

On the Jacobson radical of strongly group graded rings

Abstract: . For any non-torsion group G with identity e, we construct a strongly G-gradedring R such that the Jacobson radical J(R e ) is locally nilpotent, but J(R) is not locallynilpotent. This answers a question posed by Puczyl owski.Keywords: strongly graded rings, radicals, local nilpotencyClassification: Primary 16A03; Secondary 16A20 Several interesting results of ring theory establish the local nilpotency of theJacobson radical of some ring constructions (cf. [9]). In this paper we consideran analogous question for strongly group graded rings. Let G be a group. Anassociative ring R =M g∈G R g is said to be stronglyG-gradedif R g R h = R gh forall g,h ∈ G. Strongly group graded rings have been intensively investigated forseveral years (cf., for example, [12],[15],[20]). In [18] the following question wasposed: is it true that for every free group G of rank ≥ 2 the Jacobson radicalof each strongly G-graded ring is locally nilpotent? (As it is noted in [18], thequestion is also connected with [14], Problem 24, and with a problem on the localnilpotency of the Jacobson radical of a skew polynomial ring, cf. [19].) It followsfrom the results of [6] that the answer is positive in the case when R